Flutter boundary

Aeroelastic and vibration control technology allows flight vehicles to operate beyond the traditional flutter boundaries, improves ride qualities, and minimizes vibration fatigue damage. Conventional active flutter and vibration control technology relies on the use of aerodynamic control surfaces operated by servo-hydraulic actuators. In this conventional configuration, the... 

The variation of the real and imaginary parts of eigenvalues (i.e., natural frequencies) for = 0.218 > 0.178 as kc is varied is plotted in Fig. 7.4. Mode coupling instability occurs at the flutter boundary kc = 9.65 x 10. It is interesting to note that, due to the activation of (7.21) inequality, the origin remains unstable even when kc is large enough that the two modes decouple. 

Remark 7.6. Because of the special stmcture of the inertia matrix of the linear system (7.8), in addition to the flutter boundary, (7.49), there exist a secondary instability boundary defined by (7.48). This additional boundary corresponds to the... 

Instead of the numerical factor 4.0 in Equation 7.10, hydrodynamic theory predicts a factor near 6.0 for the effective boundary layer thickness adjacent to a flat plate (both numbers increase about 3% per 10°C Schlichting and Gersten, 2003). However, wind tunnel measurements under an appropriate turbulence intensity, as well as field measurements, indicate that 4.0 is more suitable for leaves. This divergence from theory relates to the relatively small size of leaves, their irregular shape, leaf curl, leaf flutter, and, most important, the high turbulence intensity under field conditions. Moreover, the dependency of 6bl on /° 5, which applies to large flat surfaces, does... 

When the wind velocity is increased up to a certain point the test function T will vanish, such that at least one of the roots of the characteristic equation is imaginary. Hence, T 0 is the stability boundary beyond which the bridge becomes unstable. Such a wind speed is referred to as the flutter velocity and the magnitude of the imaginary root is the flutter frequency. 

The flutter velocity and frequency can be determined through an iterative procedure similar to the procedure suggested for single degree of freedom flutter. The procedure involves the characteristic equation of Eq. (61), and the stability boundary condition T — 0, i.e.,... 

At the boundary of the flutter instability, = ficr = 0.5, the two eigenvalues are identical and by further increasing the coefficient of friction, the eigenvalues become complex numbers. 

Next we add damping by setting Cx = 1.33 N s/m and = 1 N s/m. Generally, when damping is present, similar coalescence of the eigenvalues as in the undamped case is not observed." This is certainly evident from the plot of variation of the real and imaginary parts of the eigenvalues in Fig. 4.14. In this example, the critical value of the coefficient of friction (i.e., flutter instability boundary) is... 

The second instability condition, given by (7.22), represents the mode coupling (flutter) instability. The equation for the flutter instability boundary (i.e., coalescence of the two real natural frequencies) is found by replacing the less than sign with the equal sign in (7.22). After some manipulations, one finds... 

At the second boundary of flutter instability, the squared frequencies are identical and purely imaginary, i.e., coi 2 = —a > 0. Beyond this threshold, squared frequencies are different but remain purely imaginary. 

In this chapter, the mode coupling instability in the lead screw drives was studied using several multi-DOF models. It was found that the necessary conditions for the mode coupling instability to occur are (a) the lead screw must be self-locking (i.e., p > tan 1) and (b) the direction of the applied axial force must be the same as the direction of motion of the translating part (i.e., RQ. > 0). The flutter instability boundary in the space of system parameters for the 2-DOF models of Sects. 5.5 and 5.6 was given by (7.36) and (7.49), respectively. 

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